Second Law of Newton Formula Derivation

Let’s derive the 2nd law of motion.

The rate of change in the momentum of a body is directly proportional to the applied force and occurs in the force’s direction. Newton’s first describes force in a practical way, while Newton’s second law provides a numerical explanation of force.

Consider a body with instantaneous velocity [Tex]\vec{v}        [/Tex], and momentum[Tex] \vec{p}       [/Tex] given by:

[Tex]\vec{p} = m\vec{v} [/Tex]

Since, according to the second law of motion,

[Tex]\vec{F}∝\dfrac{d\vec{p}}{dt} [/Tex]

Where [Tex]\vec{F}       [/Tex] is the force acting on the object.

Also, since the momentum is defined as,

[Tex]\vec{p} = m\vec{v} [/Tex]

Therefore, the previous equation becomes,

[Tex]\vec{F}\alpha \dfrac{d(m\vec{v})}{dt} [/Tex]

[Tex]\vec{F}=k\dfrac{d(m\vec{v})}{dt} [/Tex]

Where k is the constant of proportionality.

As the mass m of a body can be considered to be a constant quantity so derivative is applicable to the velocity of the body as shown below,

[Tex]\vec{F}=km\dfrac{d(\vec{v})}{dt} [/Tex]

It is known that the time rate change of velocity of the body is termed as its acceleration i.e.

[Tex]\vec{a}=\dfrac{d\vec{v}}{dt} [/Tex]

Therefore, 

[Tex]\vec{F}=km\vec{a} [/Tex]

The units of force are also chosen such that ‘k’ equals one.

As a result, if a unit force is selected to be the force causing a unit acceleration in a unit mass, i.e., 

F = 1 N, m = 1 kg and a = 1 ms^-2. This implies, k = 1.

Thus, Newton’s second law of motion in mathematical form is given as

[Tex]\bold{\vec{F}\ =m\vec{a}} [/Tex]

That is, the applied force of a body is defined as the product of its mass and acceleration. Hence, this provides us with a measure of the force.

If F = 0, we get a = 0. 

This is similar to Newton’s first law of motion. That is, if there is no net external force, there will be no change in state of motion, implying that its acceleration is zero.

The image given below shows a car of mass ‘M’ accelerating at ‘a’ when the force applied is F.

Newton’s Second Law Formula for Variable Mass

Let us assume a body to be at an initial point (0) specified at location L₀ and at the time instance t₀. Let us assume the body has mass m₀ and travels with a uniform velocity v₀. After applying a force of F, the object reaches its destination (point 1) at location L₁. This happens at a specific time, t₁.

The mass and velocity of the body undergo a transformation as the body travels to v₁. Deriving the values for m₁ and v_1, we get, 

[Tex]\bold{F=\dfrac{m_1v_1-m_0v_0}{t_1-t_0}} [/Tex]

Imagine a car at two different points in time as shown in below image. At the first point (marked T0), it’s moving at a certain speed (V0) and has covered a specific distance (L0). Then, at a later point (T1), its speed has changed to V1 and it has traveled a total distance of L1.

Newton’s Second Law Formula for Constant Mass

For a constant mass, the usage of Newton’s second law can be equated as follows:

[Tex]\bold{F=m\dfrac{v_1-v_0}{t_1-t_0}} [/Tex]

Derivation of Momentum

Momentum is the quantity of motion, which is the product of a body’s mass and velocity. When you walk, run, etc., you have momentum.

The momentum of a body is the product of the mass of the body and its associated velocity. Momentum can be considered to be a vector quantity, that is, it has both an associated magnitude and direction.  

Mathematically, the momentum (p) of an object of mass (m) moving with velocity (v) is defined as:

p= m × v

The unit of momentum is kg ms^-1 and its dimensional formula is [MLT^-1]

The second law of motion gives us a method to calculate the force required to move an object.

Newton’s Second Law of Motion says that for an object under the influence of unbalanced forces, the acceleration of the object is directly proportional to the force applied.

In this article, we will learn about the Second law of motion by Newton, including its definition, example, formula, derivation, and applications. We will also explore some numerical problems and FAQs on the Second Law of motion.